On the Limit of Superposition States

Abstract

In this paper, we study the structure of a family of superposition states on tensor algebras. The correlation functions of the considered states are described through a new kind of positive definite kernels valued in the dual of C${}^{\ast }$-algebras, so-called Schur kernels. Mainly, we show the existence of the limiting state of a net of superposition states over an arbitrary locally finite graph. Furthermore, we show that this limiting state enjoys a mixing property and an $\alpha$-mixing property in the case of the multi-dimensional integer lattice ${\mathbb{Z}}^{\nu }$.